motion in central-force field

Let us consider a body with m in a gravitational force field (http://planetmath.org/VectorField) exerted by the origin and directed always from the body towards the origin. Set the plane through the origin and the velocity vectorv→ of the body. Apparently, the body is forced to move constantly in this plane, i.e. there is a question of a planar motion. We want to derive the trajectory of the body.

where r→ 0 and s→ 0 are the unit vectors in the direction of r→ and of r→ rotated 90 degrees anticlockwise (r→ 0=i→⁢cos⁡φ+j→⁢sin⁡φ, whence r→ 0d⁢t=(-i→⁢sin⁡φ+j→⁢cos⁡φ)⁢d⁢φd⁢t=d⁢φd⁢t⁢s→ 0). Thus the kinetic energy of the body is

Ek=12⁢m⁢|d⁢r→d⁢t|2=12⁢m⁢((d⁢rd⁢t)2+(r⁢d⁢φd⁢t)2).

Because the gravitational force on the body is exerted along the position vector, its moment is 0 and therefore the angular momentum

This means that ψ=C-φ, where the constant C is determined by the initial conditions. We can then solve r from the first of the equations (3), obtaining

r=G2k⁢m⁢(1-G⁢qk⁢cos⁡(C-φ))=p1-ε⁢cos⁡(φ-C),

(4)

where

p:=G2k⁢m,ε:=G⁢qk.

By the http://planetmath.org/node/11724parent entry, the result (4) shows that the trajectory of the body in the gravitational field (http://planetmath.org/VectorField) of one point-like sink is always a conic section whose focus the sink causing the field.

As for the of the conic, the most interesting one is an ellipse. It occurs, by the
http://planetmath.org/node/11724parent entry, when ε<1. This condition is easily seen to be equivalent with a negative total energy E of the body.

One can say that any planet revolves around the Sun along an ellipse having the Sun in one of its foci — this is Kepler’s first law.